Schrodinger Operators on Graphs

TL;DR

This paper extends the study of discrete Schrödinger operators on graphs, introducing a symplectic form on solutions and exploring applications in scattering theory, with new methods for diagonalization of quadratic forms.

Contribution

It introduces a symplectic Wronskian on the space of solutions and applies it to scattering theory on graphs, providing new tools for analyzing these operators.

Findings

Constructed a vector-valued symplectic form on solution space.

Applied the symplectic form to scattering theory on graphs with tails.

Presented an effective diagonalization method for Fermionic quadratic forms.

Abstract

This work is the direct continuation of the author's note published in Russian Math Surveys 52 (1997), no 6. Discrete Schrodinger operators on graphs and higher dimensional simplicial complexes are considered. A vector-valued symplectic form on the space of solutions is consructed. This form, "Symplectic Wronskian", takes value in the group of 1-dimensional cycles. This construction has important applications for the Scattering Theory on graphs with tails. Effective diagonalization of the real Fermionic quadratic form is presented in the Appendix. This construction appeared first time in 1987 in the author's paper dedicated to an analogue of Morse theory for vector fields (it was published as an Appendix to the author's joint paper with M.Shubin, Soviet Math Dokl 34 (1987) no 1).